DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Cumulative coordinate of a point density on a compact interval

Definition

Let $a < b$ and let $g : [a, b] \to (0, \infty)$ be a continuously differentiable function with $\int_{a}^{b} g (t) d t = 1$ . Its cumulative point-density coordinate is

$G_{g} (x) : = \int_{a}^{x} g (t) d t, x \in [a, b] .$

The integrals here are the ordinary Riemann integrals of continuous functions. The supplier Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ guarantees that they are defined and that $G_{g}$ is a primitive of $g$ .

Depends on

Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_a^b f = G(b)-G(a)$ for any primitive $G$

Used by

Positive C1 compact one-dimensional compander Definition
A positive C1 point density has a C2 cumulative inverse Lemma

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Robert M. Gray, Source Coding Theory, first edition (standard reference, not scraped)
Robert M. Gray and David L. Neuhoff, Quantization, IEEE Transactions on Information Theory 44(6) (1998) (standard reference, not scraped)